CFA Level III · Level III Core

Portfolio Performance Evaluation: formula sheet

Full chapter guide

Key formulas

Active return: Active return = Rp − Rb; Portfolio return minus benchmark return. This is the total that attribution must explain.
Allocation effect (one segment, Brinson-style): Allocation = (wp − wb) × (Rb,i − Rb); Active weight in a segment times the segment's benchmark return relative to the total benchmark return. Rewards overweighting segments that beat the overall benchmark.
Selection effect (one segment): Selection = wb × (Rp,i − Rb,i); Benchmark weight times the segment return difference. Measures security choice within the segment.
Interaction effect (one segment): Interaction = (wp − wb) × (Rp,i − Rb,i); Combined effect of active weight and active selection. Some methods merge it into selection.
Sum check: Allocation + Selection + Interaction (summed over segments) = Rp − Rb; Holds when weights each sum to 1 and segment returns aggregate to total returns. Use it to check your work.
Active return: Active return = Rp − Rb; Rp is total portfolio return, Rb is total benchmark return. Attribution effects must sum to this.
Allocation effect (sector i): Allocation_i = (wp,i − wb,i) × (Rb,i − Rb); Subtract the total benchmark return, not zero. This is the common Brinson-Fachler form. Some texts use (wp,i − wb,i) × Rb,i, so follow the form given in the question.
Selection effect (sector i): Selection_i = wb,i × (Rp,i − Rb,i); Uses the benchmark weight, so it isolates stock picking from weight differences.
Interaction effect (sector i): Interaction_i = (wp,i − wb,i) × (Rp,i − Rb,i); Joint effect of weight and return differences. Sometimes combined with selection.
Total sector contribution: Allocation_i + Selection_i + Interaction_i = wp,i × Rp,i − wb,i × Rb,i − (wp,i − wb,i) × Rb; Summing over all sectors gives Rp − Rb.
Factor model active return: Active return = Σ (active exposure_k × factor return_k) + security selection (residual); Active exposure is portfolio factor exposure minus benchmark exposure.
Macro attribution manager effect: Manager value added = Manager return − Manager's benchmark return; Sponsor-level results also compare the actual fund with the policy benchmark, with differences traced to allocation away from policy and to manager returns.
Sharpe ratio: Sharpe = (Rp − Rf) ÷ σp; Rp is portfolio return, Rf the risk-free rate, σp the portfolio standard deviation. Use for total risk.
Treynor ratio: Treynor = (Rp − Rf) ÷ βp; Use for systematic risk, mainly for a portfolio that is part of a diversified whole.
M-squared: M² = Rf + Sharpe_p × σB − RB, equivalently (Rp − Rf) × (σB ÷ σp) + Rf − RB; σB is the benchmark standard deviation. Positive means the portfolio beat the benchmark at equal total risk. Some texts quote the risk-matched return Rf + Sharpe_p × σB before subtracting RB; read the question.
Jensen's alpha: α = Rp − [Rf + βp × (Rm − Rf)]; Rm is the market (or benchmark proxy) return. Alpha is the return above what CAPM predicts for the portfolio's beta. Positive alpha means outperformance.
Information ratio: IR = (Rp − RB) ÷ tracking risk; Tracking risk is the standard deviation of active returns (Rp − RB) over time.
Sortino ratio: Sortino = (Rp − MAR) ÷ downside deviation; MAR is the minimum acceptable return. Downside deviation counts only returns below MAR.
Active return: Active return = Rp − Rb; Portfolio return minus benchmark return over the same period.
Tracking error: Tracking error = standard deviation of (Rp − Rb); Measures how closely the portfolio follows the benchmark. Tracking error well above the manager's intended active risk may reflect a poor benchmark fit or extra active risk taken by the manager.
Systematic bias test: Rp = a + b × Rb + error, or equivalently (Rp − Rb) = a + (b − 1) × Rb + error; Regress portfolio return on benchmark return: a good benchmark has b close to 1 and a close to 0. In the active return form, the slope (b − 1) should be close to 0. A slope clearly away from 0 signals benchmark bias.
Information ratio: IR = average active return ÷ tracking error; Uses the benchmark as the reference, so a poor benchmark distorts it.
Benchmark properties checklist: Specified in advance, Appropriate, Measurable, Unambiguous, Reflective of current opinions, Accountable, Investable (SAMURAI); A memory aid for the seven properties of a valid benchmark.
Active return: Active return = R(portfolio) − R(benchmark); Calculate it each period, then take the mean and the standard deviation.
Information ratio: IR = mean active return ÷ tracking risk; Tracking risk is the standard deviation of active returns. Keep both on the same time basis, for example annualised.
Up-capture ratio: Up capture = manager's average return in up-benchmark periods ÷ benchmark's average return in those periods; Above 100% is desirable. Use only periods when the benchmark return is positive.
Down-capture ratio: Down capture = manager's average return in down-benchmark periods ÷ benchmark's average return in those periods; Below 100% is desirable, since the manager loses less than the benchmark.
Capture ratio: Capture ratio = up capture ÷ down capture; Above 1 suggests favourable asymmetry.
t-statistic of mean active return: t = mean active return ÷ (tracking risk ÷ √n); n is the number of periods. Compare with a critical value using n − 1 degrees of freedom. t = IR × √n only when the IR is calculated on the same period basis as n (for example, annual IR with n years).
Returns-based style analysis: R(p) = w1·R(index 1) + … + wk·R(index k) + e, with wi ≥ 0 and Σwi = 1; The weights show effective style. The residual e is the selection return, the part not explained by the style indexes.

Quick revision

Active return = portfolio return − benchmark return.
Attribution splits active return into allocation, selection and interaction effects.
Macro attribution looks at the sponsor's decisions across managers or asset classes; micro attribution looks at the manager's decisions.
Sharpe ratio = (Rp − Rf) ÷ σp, using total risk.
Treynor ratio = (Rp − Rf) ÷ βp, using systematic risk.
M² restates the portfolio's return at the market's total risk so it is in return units.
Information ratio = active return ÷ tracking risk.
Jensen's alpha = Rp − [Rf + βp(Rm − Rf)].
A valid benchmark is specified in advance, appropriate, measurable, unambiguous, reflective of current investment views, accountable and investable.
Style analysis regresses returns on style indexes to find the manager's actual style and any drift.
Choose the measure to fit the case: Sharpe for a whole portfolio, Treynor or alpha for one part of a diversified portfolio.
Show every calculation step in essays and give exactly the number of responses asked for.

Common mistakes

Using segment benchmark return alone in the allocation effect. Fix: Subtract the total benchmark return: (wp − wb) × (Rb,i − Rb). Otherwise an overweight in any positive-return segment looks like skill.
Treating attribution as proof of skill. Fix: Remember that attribution describes sources. Appraisal tests skill using risk adjustment and consistency.
Using the benchmark sector return alone in the allocation formula without subtracting the total benchmark return. Fix: Use (wp − wb) × (Rb,i − Rb) unless the question states otherwise. Weight differences sum to zero, so subtracting the total return makes allocation measure sector bets against the benchmark average.
Using portfolio weight instead of benchmark weight in the selection effect. Fix: Selection uses wb. The portfolio-weight part of the gap belongs to interaction.
Using Sharpe for a sleeve inside a diversified portfolio, or Treynor for a whole portfolio. Fix: Ask first whether total or systematic risk matters to the investor, then pick the measure.
Subtracting the wrong reference return in the numerator. Fix: Sharpe and Treynor subtract Rf in the numerator. Jensen's alpha subtracts the CAPM-required return. The information ratio subtracts the benchmark return.
Treating a manager universe as a valid benchmark. Fix: Remember that universes are not unambiguous, not investable and not specified in advance, and they suffer from survivorship and classification bias.
Using a broad market index for a style-specific manager. Fix: Match the benchmark to the style. A small-cap value manager measured against a large-cap index shows misleading active return.
Calling a positive active return proof of skill Fix: Always check significance with the t-statistic and sample length before saying skill.
Using total standard deviation in the information ratio Fix: IR uses tracking risk, the standard deviation of active returns, in the denominator.

Exam tips

Match the command word: calculate means show the number, identify means name the source, justify means give a reason tied to the mandate.
Always show the sum check. In essay calculation items, a correct number typed on its own earns full credit, but showing your working still helps you check your answer and catch errors.
Keep the three components distinct in written answers: measurement is what, attribution is why, appraisal is skill and risk.
In essay sets, give only the number of responses requested, in order, and keep each to a short point linked to the client's objectives and benchmark.
Read the command word. 'Calculate' needs a number on its own line. 'Explain' or 'justify' needs a short reason tied to the vignette, such as which decision added value.
Show the grid and each formula even if the answer is a single number, so a slip in one step does not cost the whole item.
Always run the sum check against Rp − Rb. It catches most errors in seconds.
Interpretation items often ask which effect shows skill. Link selection to security picking and allocation to sector or asset class weighting, and remember a factor model's residual is the security-specific part.